it is not necessary for us to do more than sketch in the most cursory manner what is called the Method of Least Squares. 341. Supposing the zero-point and the graduation of an instrument (micrometer, mural circle, thermometer, electrometer, galvanometer, etc.) to be absolutely accurate, successive readings of the value of a quantity (linear distance, altitude of a star, temperature, potential, strength of an electric current, etc.) may, and in general do, continually differ. What is most probably the true value of the observed quantity? The most probable value, in all such cases, if the observations are all equally reliable, will evidently be the simple mean; or if they are not equally reliable, the mean found by attributing weights to the several observations in proportion to their presumed exactness. But if several such means have been taken, or several single observations, and if these several means or observations have been differently qualified for the determination of the sought quantity (some of them being likely to give a more exact value than others), we must assign theoretically the best method of combining them in practice. 342. Inaccuracies of observation are, in general, as likely to be in excess as in defect. They are also (as before observed) more likely to be small than great; and (practically) large errors are not to be expected at all, as such would come under the class of avoidable mistakes. It follows that in any one of a series of observations of the same quantity the probability of an error of magnitude x, must depend upon x, and must be expressed by some function whose value diminishes very rapidly as increases. The probability that the error lies between x and x+dx, where dx is very small, must also be proportional to dx. The law of error thus found is where is a constant, indicating the degree of coarseness or delicacy I secures of the system of measurement employed. The co-efficient that the sum of the probabilities of all possible errors shall be unity, as it ought to be. 343. The Probable Error of an observation is a numerical quantity such that the error of the observation is as likely to exceed as to fall short of it in magnitude. If we assume the law of error just found, and call P the probable error in one trial, we have the approximate result P=0'477 h. 344. The probable error of any given multiple of the value of an observed quantity is evidently the same multiple of the probable error of the quantity itself. The probable error of the sum or difference of two quantities, affected by independent errors, is the square root of the sum of the squares of their separate probable errors. 345. As above remarked, the principal use of this theory is in the deduction, from a large series of observations, of the values of the quantities sought in such a form as to be liable to the smallest probable error. As an instance-by the principles of physical astronomy, the place of a planet is calculated from assumed values of the elements of its orbit, and tabulated in the Nautical Almanac. The observed places do not exactly agree with the predicted places, for two reasons' -first, the data for calculation are not exact (and in fact the main object of the observation is to correct their assumed values); second, the observation is in error to some unknown amount. Now the difference between the observed, and the calculated, places depends on the errors of assumed elements and of observation. Our methods are applied to eliminate as far as possible the second of these, and the resulting equations give the required corrections of the elements. Thus if be the calculated R.A. of a planet: da, de, dw, etc., the corrections required for the assumed elements: the true R.A. is θ + Αδα + δε + Πδα + etc., where A, E, II, etc., are approximately known. Suppose the observed R.A. to be , then a known quantity, subject to error of observation. Every observation made gives us an equation of the same form as this, and in general the number of observations greatly exceeds that of the quantities da, Se, Sw, etc., to be found. 346. The theorems of § 344 lead to the following rule for combining any number of such equations which contain a smaller number of unknown quantities :- Make the probable error of the second member the same in each equation, by the employment of a proper factor: multiply each equation by the co-efficient of x in it and add all, for one of the final equations; and so, with reference to y, z, etc., for the others. The probable errors of the values of x, y, etc., found from these final equations will be less than those of the values derived from any other linear method of combining the equations. This process has been called the method of Least Squares, because the values of the unknown quantities found by it are such as to render the sum of the squares of the errors of the original equations a minimum. 347. When a series of observations of the same quantity has been made at different times, or under different circumstances, the law connecting the value of the quantity with the time, or some other variable, may be derived from the results in several ways-all more or less approximate. Two of these methods, however, are so much more extensively used than the others, that we shall devote a page or two here to a preliminary notice of them, leaving detailed instances of their application till we come to Heat, Electricity, etc. They consist in (1) a Curve, giving a graphic representation of the relation between the ordinate and abscissa, and (2) an Empirical Formula connecting the variables. 348. Thus if the abscissae represent intervals of time, and the ordinates the corresponding height of the barometer, we may construct curves which show at a glance the dependence of barometric pressure upon the time of day; and so on. Such curves may be accurately drawn by photographic processes on a sheet of sensitive paper placed behind the mercurial column, and made to move past it with a uniform horizontal velocity by clockwork. A similar process is applied to the Temperature and Electricity of the atmosphere, and to the components of terrestrial magnetism. 349. When the observations are not, as in the last section, continuous, they give us only a series of points in the curve, from which, however, we may in general approximate very closely to the result of continuous observation by drawing, liberâ manu, a curve passing through these points. This process, however, must be employed with great caution; because, unless the observations are sufficiently close to each other, most important fluctuations in the curve may escape notice. It is applicable, with abundant accuracy, to all cases where the quantity observed changes very slowly. Thus, for instance, weekly observations of the temperature at depths of from 6 to 24 feet underground were found by Forbes sufficient for a very accurate approximation to the law of the phenomenon. 350. As an instance of the processes employed for obtaining an empirical formula, we may mention methods of Interpolation, to which the problem can always be reduced. Thus from sextant observations, at known intervals, of the altitude of the sun, it is a common problem of Astronomy to determine at what instant the altitude is greatest, and what is that greatest altitude. The first enables us to find the true solar time at the place, and the second, by the help of the Nautical Almanac, gives the latitude. The calculus of finite differences gives us formulae proper for various data; and Lagrange has shown how to obtain a very useful one by elementary algebra. In finite differences we have f(x+h) = f(x)+ h▲ƒ (x) + h (h− 1) f(x) + ... I.2 This is useful, inasmuch as the successive differences, Af(x), Af(x), etc., are easily calculated from the tabulated results of observation, provided these have been taken for equal successive increments of x. If for values x, x,,..., a function takes the values y1, y', y„....... yn, Lagrange gives for it the obvious expression Here is assumed that the function required is a rational and integral one in x of the n-1th degree; and, in general, a similar limitation is in practice applied to the other formula above; for in order to find the complete expression for f(x), it is necessary to determine the values of ▲ƒ (x), ▲3ƒ (x),.......... If n of the co-efficients be required, so as to give the n chief terms of the general value of ƒ (x), we must have n observed simultaneous values of x and ƒ(x), and the expression becomes determinate and of the n-1th degree in h. In practice it is usually sufficient to employ at most three terms of the first series. Thus to express the léngth / of a rod of metal as depending on its temperature t, we may assume 1=1+A(t-t)+B(t−t), 7 being the measured length at any temperature to A and B are to be found by the method of least squares from values of / observed for different given values of t. 351. These formulae are practically useful for calculating the probable values of any observed element, for values of the independent variable lying within the range for which observation has given values of the element. But except for values of the independent variable either actually within this range, or not far beyond it in either direction, these formulae express functions which, in general, will differ more and more widely from the truth the further their application is pushed beyond the range of observation. In a large class of investigations the observed element is in its nature a periodic function of the independent variable. The harmonic analysis (§ 88) is suitable for all such. When the values of the independent variable for which the element has been observed are not equidifferent the co-efficients, determined according to the method of least squares, are found by a process which is necessarily very laborious; but when they are equidifferent, and especially when the difference is a submultiple of the period, the equation derived from the method of least squares becomes greatly simplified. Thus, if ℗ denote an angle increasing in proportion to t, the time, through four right angles in the period, T, of the phenomenon; so that let f(0) = A + 4, cos 0 + A, cos 20+ ... +B, sin 0 + B, sin 20+ where A A, A,...B, B,... are unknown co-efficients, to be determined so that f(0) may express the most probable value of the element, not merely at times between observations, but through all time as long as the phenomenon is strictly periodic. By taking as many of these coefficients as there are of distinct data by observation, the formula is made to agree precisely with these data. But in most applications of the method, the periodically recurring part of the phenomenon is expressible by a small number of terms of the harmonic series, and the higher terms, calculated from à great number of data, express either irregularities of the phenomenon not likely to recur or errors of observation. Thus a comparatively small number of terms may give values of the element even for the very times of observation, more probable than the values actually recorded as having been observed, if the observations are numerous but not minutely accurate. 2π The student may exercise himself in writing out the equations to determine five, or seven, or more of the coefficients according to the method of least squares; and reducing them by proper formulae of analytical trigonometry to their simplest and most easily calculated forms where the values of 0 for which ƒ(0) is given are equidifferent. He will thus see that when the difference is ¿ being any integer, and when the number of the data is i or any multiple of it, the equations contain each of them only one of the unknown quantities: so that the method of least squares affords the most probable values of the co-efficients, by the easiest and most direct elimination. |